\begin{answer}
We have

$$
\begin{aligned}
D_{KL }(p_\theta\|p_{\tilde \theta})&\approx D_{KL}(p_\theta\| p_{\theta}) + (\tilde \theta - \theta)^T\nabla_{\theta'}\log p(y;\theta')|_{\theta' = \theta} + \frac{1}{2}(\tilde \theta - \theta)^T\nabla_{\theta'}^2\log p(y;\theta')|_{\theta' = \theta}
(\tilde \theta - \theta)\\
&= 0 + (\tilde \theta - \theta)^T E_{y\sim p(y;\theta)}[-\nabla_{\theta'}\log p(y;\theta')|_{\theta' = \theta}] + \frac{1}{2}(\tilde \theta - \theta)^T E_{y\sim p(y;\theta)}[-\nabla_{\theta'}^2\log p(y;\theta')|_{\theta' = \theta}](\tilde \theta - \theta)\\
&= \frac{1}{2}d^T\mathcal I(\theta) d
\end{aligned}
$$
\end{answer}
